# Cutting towers of number fields

## TL;DR

This paper explores the structure of maximal pro-$p$ extensions of number fields unramified outside a finite set of places, constructing infinite subextensions with bounded ramification and improving bounds on root discriminants.

## Contribution

It introduces refined Golod-Shafarevich criteria to construct infinite subextensions with bounded ramification, achieving new records on Martinet constants and answering Ihara's question.

## Key findings

- Constructed infinite subextensions with bounded ramification.
- Achieved new records on Martinet constants in specific cases.
- Produced infinite asymptotically good extensions with infinitely many split primes.

## Abstract

Given a prime $p$, a number field $\K$ and a finite set of places $S$ of $\K$, let $\K_S$ be the maximal pro-$p$ extension of $\K$ unramified outside $S$. Using the Golod-Shafarevich criterion one can often show that $\K_S/\K$ is infinite. In both the tame and wild cases we construct infinite subextensions with bounded ramification using the refined Golod-Shafarevich criterion. In the tame setting we achieve new records on Martinet constants (root discriminant bounds) in the totally real and totally complex cases. We are also able to answer a question of Ihara by producing infinite asymptotically good extensions in which infinitely many primes split completely.